HARMONICPHASE · New perspectives in phase-space analysis and Fourier restriction

The primary objective of this project is to make progress on central questions in Harmonic Analysis and related areas, notably Fourier restriction/extension theory and phase-space Analysis, through the lens of new phase-space representations involving the Wigner distribution.Fourier restriction/extension theory — a driving force of modern Harmonic Analysis — aims to effectively control the constructive interference of bundles of travelling waves. Several physical phenomena are described in terms of Fourier extension operators. In simple terms, these objects convert input functions — associated to the initial data of a partial differential equation (PDE) — into bundles of plane waves travelling in different directions. The Schrödinger and Helmholtz equations are the most well-known examples of PDEs whose solutions are given by Fourier extension operators. The former describes how the wave function of a quantum system evolves, whereas the latter models, for instance, the behaviour of electromagnetic waves in confined spaces like optical fibres.Energy maximisation is a modern challenge in Signal Processing and phase-space Analysis, with applications in medical imaging resolution, noise reduction in audio processing and seismology, to mention a few. An exciting link between Fourier extension operators and the problem of energy maximisation was recently discovered in my recent work via Flandrin's conjecture, which predicts that convex sets are efficient phase-space signal filters: isolating the phase-space content of a signal to a convex set via its Wigner distribution does not increase the signal's energy by too much.By capitalising on the phase-space perspective provided by Wigner-type distributions, this highly intradisciplinary project aims to establish new results and connections in two interrelated programmes covering both areas described above.